Abstract
Cyclic codes have efficient encoding and decoding algorithms. The decoding error probability and the undetected error probability are usually bounded by or given from the weight distributions of the codes. Most researches are about the determination of the weight distributions of cyclic codes with few nonzeros, by using quadratic forms and exponential sums but limited to low moments. In this paper, we focus on the application of higher moments of the exponential sums to determine the weight distributions of a class of ternary cyclic codes with three nonzeros, combining with not only quadratic forms but also MacWilliams’ identities. Another application of this paper is to emphasize the computer algebra system Magma for the investigation of the higher moments. In the end, the result is verified by one example using Matlab.
Highlights
Cyclic codes have a lot of applications in communication system, storage system, and computers, and they have been studied for a long time [1, 2]
We focus on the application of higher moments of the exponential sums to determine the weight distributions of a class of ternary cyclic codes with three nonzeros, combining with quadratic forms and MacWilliams’ identities
We focus on the application of higher moments of the exponential sums to determine the weight distributions of a class of ternary cyclic codes with three nonzeros, combining with quadratic forms and MacWilliams’ identities, with the help of the computer algebra system Magma
Summary
Cyclic codes have a lot of applications in communication system, storage system, and computers, and they have been studied for a long time [1, 2]. As shown in [8, 9], the problem of computing weight distribution of a cyclic code is connected with the evaluation of certain exponential sums, which are generally hard to be determined explicitly. We focus on the application of higher moments of the exponential sums to determine the weight distributions of a class of ternary cyclic codes with three nonzeros, combining with quadratic forms and MacWilliams’ identities, with the help of the computer algebra system Magma. An [n, k] linear code C over Fp is called cyclic if This paper is the counterpart of our other result in [19]
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